= Solution
Let $E_a$ denote the frame dual to the coframe. For the convention just stated, the <Ricci tensor> is obtained from the <curvature 2-forms> through the Ricci one-forms $\mathcal R_b=\sum_a\iota_{E_a}\Theta^a{}_b=R_{bd}e^d$. In particular $\iota_{E_1}(\ell\wedge e^1)=-\ell$ and likewise for index two. It follows that
$$
\mathcal R_0=-(P+Q)\ell,\qquad \mathcal R_3=(P+Q)\ell,\qquad \mathcal R_1=\mathcal R_2=0.
$$
Thus
$$
\boxed{\operatorname{Ric}=-(P+Q)\,du\otimes du,\qquad R_{tt}=R_{zz}=-(P+Q),\quad R_{tz}=R_{zt}=P+Q.}
$$
The <scalar curvature> vanishes even before imposing vacuum because $du$ is null, but the full tensor vanishes exactly when
$$
\boxed{\frac{A''}{A}+\frac{B''}{B}=0.}
$$
This is the <Vacuum Einstein equations> in this family. The nonzero-coframe hypothesis makes division by $A$ and $B$ legitimate. A nonzero $P=-Q$ can leave the <Riemann curvature tensor> nonzero even in vacuum.
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